Solving 1.6 * (x-3)/(x+3) - (x+3)/(x^2-9) A Step-by-Step Guide
Introduction: Diving into Algebraic Expressions
In the realm of mathematics, algebraic expressions often present themselves as intricate puzzles, demanding careful manipulation and a keen understanding of fundamental principles. This article delves into the process of solving a specific equation: 1.6 * (x-3)/(x+3) - (x+3)/(x^2-9). We will embark on a step-by-step journey, demystifying each operation and illuminating the path to the solution. Our primary focus is to ensure that readers, regardless of their mathematical background, can grasp the core concepts and apply them to similar problems. We'll start by identifying the key components of the equation, simplifying complex fractions, and ultimately isolating the variable x. This journey will not only provide the solution to this particular equation but also equip you with a versatile toolkit for tackling a wide array of algebraic challenges. Understanding how to simplify rational expressions is crucial in many areas of mathematics, including calculus and advanced algebra. Furthermore, the techniques we will employ, such as finding common denominators and factoring quadratic expressions, are foundational skills that are repeatedly used in various mathematical contexts. The goal is to transform a seemingly daunting equation into a manageable problem, showcasing the elegance and power of algebraic manipulation.
Deconstructing the Equation: Initial Assessment
Before we plunge into the mechanics of solving, it's vital to dissect the equation and identify its key features. The equation at hand is 1.6 * (x-3)/(x+3) - (x+3)/(x^2-9). We observe a few critical elements: a decimal coefficient (1.6), rational expressions (fractions with polynomials in the numerator and denominator), and a quadratic expression (x^2-9). Each of these elements necessitates a specific approach. The decimal coefficient suggests we might want to convert it into a fraction to simplify calculations. The rational expressions require us to consider common denominators and potential simplifications. The quadratic expression hints at the possibility of factoring. Factoring the quadratic expression is often a crucial step because it can reveal common factors with other parts of the equation, leading to significant simplifications. In this case, recognizing that x^2 - 9 is a difference of squares will be pivotal in streamlining the solution process. Moreover, it is essential to identify any restrictions on the variable x. Since we have rational expressions, we must ensure that the denominators do not equal zero. This will lead us to exclude certain values of x from our solution set. Neglecting these restrictions can lead to extraneous solutions, which are solutions obtained through algebraic manipulation but do not satisfy the original equation. Therefore, a thorough initial assessment sets the stage for an accurate and efficient solution.
Laying the Groundwork: Transforming 1.6 into a Fraction and Factoring
Our initial assessment revealed the presence of a decimal coefficient and a quadratic expression. Let's address these first to pave the way for simplification. Converting 1.6 into a fraction, we get 16/10, which further simplifies to 8/5. This fractional representation often makes algebraic manipulations cleaner and less prone to errors. Next, we turn our attention to the quadratic expression, x^2 - 9. Recognizing this as a difference of squares, we can factor it into (x+3)(x-3). This factorization is a cornerstone of the solution because it unveils a common factor with one of the denominators in our original equation. By factoring x^2 - 9, we immediately see the potential for canceling out terms, which is a fundamental strategy in simplifying rational expressions. This step not only reduces the complexity of the equation but also makes it easier to identify the common denominator needed to combine the fractions. The ability to recognize and apply factoring patterns, such as the difference of squares, is a crucial skill in algebra. It allows us to transform complex expressions into simpler, more manageable forms. This preliminary step of converting the decimal and factoring the quadratic expression demonstrates the importance of strategic preparation in tackling algebraic problems.
Combining Fractions: The Common Denominator
Now that we have transformed 1.6 into 8/5 and factored x^2 - 9 into (x+3)(x-3), our equation looks like this: (8/5) * (x-3)/(x+3) - (x+3)/((x+3)(x-3)). The next crucial step is to combine the two fractions. To do this, we need to find a common denominator. Observing the denominators, we see that the first fraction has (x+3) and the second has (x+3)(x-3). Therefore, the least common denominator (LCD) is (x+3)(x-3). We need to multiply the first fraction by (x-3)/(x-3) to obtain the common denominator. This gives us: (8/5) * (x-3)/(x+3) * (x-3)/(x-3) - (x+3)/((x+3)(x-3)). Simplifying the first fraction, we get: (8(x-3)^2) / (5(x+3)(x-3)) - (x+3)/((x+3)(x-3)). Now that both fractions have the same denominator, we can combine them. This process of finding a common denominator is a fundamental technique in working with rational expressions. It allows us to add and subtract fractions in a systematic way. The LCD ensures that we are working with equivalent fractions, which is essential for maintaining the integrity of the equation. Careful attention to this step is crucial because errors in finding the LCD can propagate through the rest of the solution.
Simplifying the Numerator: Expanding and Combining Terms
With the common denominator in place, we can now combine the fractions in our equation: (8(x-3)^2) / (5(x+3)(x-3)) - (x+3)/((x+3)(x-3)). Combining the numerators, we get: (8(x-3)^2 - 5(x+3)) / (5(x+3)(x-3)). The next step involves expanding the numerator and simplifying. First, we expand (x-3)^2 to get x^2 - 6x + 9. Then, we multiply this by 8 to get 8x^2 - 48x + 72. Next, we distribute the -5 to (x+3) to get -5x - 15. Now, our numerator looks like this: 8x^2 - 48x + 72 - 5x - 15. Combining like terms, we get: 8x^2 - 53x + 57. This simplification process is a crucial step in solving algebraic equations. By expanding and combining like terms, we reduce the complexity of the expression and make it easier to identify potential solutions. The ability to accurately perform these algebraic manipulations is essential for success in mathematics. Furthermore, keeping track of signs and coefficients during this process is paramount to avoid errors. A methodical approach, such as writing each step clearly, can help ensure accuracy.
The Simplified Equation: 8x^2 - 53x + 57 = 0
After expanding and simplifying the numerator, our equation has transformed into: (8x^2 - 53x + 57) / (5(x+3)(x-3)). To solve this equation, we set the numerator equal to zero: 8x^2 - 53x + 57 = 0. This is because a fraction is equal to zero only if its numerator is zero. Now we have a quadratic equation to solve. Solving quadratic equations is a fundamental skill in algebra, and there are several methods we can use, including factoring, using the quadratic formula, or completing the square. In this case, we will attempt to factor the quadratic expression. Factoring a quadratic equation involves finding two binomials that, when multiplied together, give us the original quadratic expression. If we can successfully factor the equation, we can then set each factor equal to zero and solve for x. This method is often the most efficient way to solve quadratic equations, but it requires practice and a good understanding of factoring techniques. The simplified equation, 8x^2 - 53x + 57 = 0, represents a significant milestone in our solution process, as it transforms the original complex rational equation into a more manageable quadratic equation.
Solving the Quadratic: Factoring or the Quadratic Formula
We've arrived at the quadratic equation 8x^2 - 53x + 57 = 0. Our next task is to find the values of x that satisfy this equation. One approach is to attempt to factor the quadratic expression. Factoring, if possible, is often the quickest route to a solution. We look for two binomials (ax + b)(cx + d) such that ac = 8, bd = 57, and ad + bc = -53. This can sometimes be a process of trial and error, but with practice, certain patterns become more apparent. However, if factoring proves difficult, we can turn to the quadratic formula, a reliable method for solving any quadratic equation of the form ax^2 + bx + c = 0. The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a). In our case, a = 8, b = -53, and c = 57. Plugging these values into the formula, we get: x = (53 ± √((-53)^2 - 4 * 8 * 57)) / (2 * 8). Whether we choose to factor or use the quadratic formula, the goal is the same: to find the roots of the quadratic equation, which represent the solutions for x. The quadratic formula guarantees a solution, even when factoring is not straightforward, making it a powerful tool in algebra.
Applying the Quadratic Formula: Finding the Roots
Since factoring 8x^2 - 53x + 57 might be challenging, let's employ the quadratic formula: x = (53 ± √((-53)^2 - 4 * 8 * 57)) / (2 * 8). First, we calculate the discriminant, which is the part under the square root: (-53)^2 - 4 * 8 * 57 = 2809 - 1824 = 985. Now we can substitute this back into the formula: x = (53 ± √985) / 16. This gives us two possible solutions for x: x = (53 + √985) / 16 and x = (53 - √985) / 16. We can approximate √985 as roughly 31.4. Therefore, our solutions are approximately: x ≈ (53 + 31.4) / 16 ≈ 5.28 and x ≈ (53 - 31.4) / 16 ≈ 1.35. The quadratic formula provides a systematic way to find the roots of any quadratic equation, even those that are difficult to factor. The discriminant, in particular, provides valuable information about the nature of the roots: a positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (a repeated root), and a negative discriminant indicates two complex roots. In this case, the positive discriminant confirms that we have two distinct real solutions. Approximating the square root allows us to obtain numerical values for the solutions, which can be useful in practical applications.
Checking for Extraneous Solutions: The Crucial Verification
We have found two potential solutions for x: approximately 5.28 and 1.35. However, we must remember our initial assessment, where we identified potential restrictions on x. Our original equation, 1.6 * (x-3)/(x+3) - (x+3)/(x^2-9), contains rational expressions. This means that any value of x that makes the denominator zero is not a valid solution. The denominators are (x+3) and (x^2-9), which factors to (x+3)(x-3). Therefore, x cannot be -3 or 3. Neither of our potential solutions, 5.28 and 1.35, violates these restrictions. Thus, both solutions are valid. Checking for extraneous solutions is a critical step in solving equations involving rational expressions. It ensures that the solutions we obtain through algebraic manipulation actually satisfy the original equation. Neglecting this step can lead to incorrect answers. The process involves identifying any values of the variable that would make a denominator zero and then verifying that these values are not among our solutions. This final check reinforces the importance of careful and thorough problem-solving in mathematics.
Conclusion: The Solutions and the Process
In conclusion, after a detailed step-by-step process, we have successfully solved the equation 1.6 * (x-3)/(x+3) - (x+3)/(x^2-9). We converted the decimal to a fraction, factored the quadratic expression, found a common denominator, simplified the numerator, and solved the resulting quadratic equation using the quadratic formula. Our solutions are approximately x ≈ 5.28 and x ≈ 1.35. We also verified that these solutions are not extraneous by checking for values that would make the denominators of the original equation zero. This journey through the equation highlights the importance of a systematic approach to problem-solving in algebra. Each step, from the initial assessment to the final verification, plays a crucial role in ensuring an accurate and complete solution. The techniques we employed, such as factoring, finding common denominators, and using the quadratic formula, are fundamental tools in mathematics. Moreover, the emphasis on checking for extraneous solutions underscores the need for careful attention to detail. By mastering these skills and adopting a methodical approach, you can confidently tackle a wide range of algebraic challenges. The process of solving this equation is not just about finding the answers; it's about developing a deeper understanding of algebraic principles and honing problem-solving abilities.
